Optimal $C^{1,\alpha}$ estimates for a class of elliptic quasilinear equations

TL;DR

This paper derives optimal $C^{1,eta}$ regularity estimates for solutions to a broad class of singular and degenerate quasilinear elliptic equations, including the p-Laplacian with variable coefficients, based on the regularity of coefficients and source term.

Contribution

It establishes sharp $C^{1,eta}$ estimates for weak solutions of singular and degenerate quasilinear elliptic equations, extending regularity results to variable coefficient cases with optimal exponents.

Findings

Sharp $C^{1,eta}$ regularity estimates are obtained.

The regularity exponent $eta$ is asymptotically optimal.

Results include the standard p-Laplacian as a special case.

Abstract

In this article we establish sharp $C^{1,\alpha}$ estimates for weak solutions of singular and degenerate quasilinear elliptic equation $$-\,div\, a(x, \nabla u) = f,$$ which includes the standard $p$-laplacean equation with varying coefficients as a special case. The sharp exponent $\alpha$ is asymptotically optimal and is determined by the H\"older regularity of the coefficients, the exponent $p$ and the $q$-integrability of the source term $f$.